Products of matrices $[\begin{array}{cc}1& 1\\ 0& 1\end{array}]$ and $[\begin{array}{cc}1& 0\\ 1& 1\end{array}]$ and the distribution of reduced quadratic irrationals

Abstract

Let $\Phi (N)$ denote the number of products of matrices $[\begin{array}{cc}1& 1\\ 0& 1\end{array}]$ and $[\begin{array}{cc}1& 0\\ 1& 1\end{array}]$ of trace equal to $N$, and $\Psi (N)={\sum }_{n=3}^N\Phi (n)$ be the number of such products of trace between $3$ and $N$. We prove an asymptotic formula of type $\Psi (N)=c_1{N}^2\log N+c_2{N}^2+O_{\epsilon }(N^{7/4+\epsilon })$ as $N\to \infty$. As a result, the Dirichlet series ${\sum }_{n=1}^{\infty }\Phi (n)n^{-s}$ has a meromorphic extension in the half-plane $\Re (s)>7/4$ with a single, order two pole at $s=2$. Our estimate also improves on an asymptotic result of Faivre concerning the distribution of reduced quadratic irrationals, providing an explicit upper bound for the error term.

17 pages; catches a few typos in the printed version

References in corpus (1)

• The Phase Transition in Statistical Models Defined on Farey Fractions

Cited by in corpus (1)

• Distribution of periodic points of certain Gauss shifts with infinite invariant measure